Two Lines, One Solution.

Simultaneous equations are just a fancy way of asking: “Where do these two lines cross?” Here are the two tools you need to find that X marks the spot.

The Two Contenders

When you have two equations with two unknowns (like $x$ and $y$ ), you need a strategy to solve them. You have two main weapons in your arsenal:

Substitution Method (The “Plug-In” Move)
Elimination Method (The “Cancel-Out” Move)

Method 1: The Substitution Method

Best For: When one variable is already lonely (isolated).

Example: $y = 2x + 1$ (y is alone!)
Example: $x = 3y - 5$ (x is alone!)

Example: Using Substitution

Equations:

$2x + 3y = 13$
$y = x - 4$ $\leftarrow$ Look! y is already isolated.

Step 1: Substitute Plug equation (2) into equation (1). $2x + 3(x - 4) = 13$

Step 2: Solve for x $2x + 3x - 12 = 13$ $5x = 25$ $x = 5$

Step 3: Solve for y Plug $x = 5$ back into equation (2). $y = (5) - 4$ $y = 1$

Solution: $x = 5, y = 1$

Method 2: The Elimination Method

Best For: When variables have the same (or easy to match) coefficients.

Example: $3x + 2y = 10$ and $3x - y = 5$ (3x matches!)

Example: Using Elimination

Equations:

$5x + 2y = 20$
$3x - 2y = 12$ $\leftarrow$ Look! +2y and -2y are perfect opposites.

Step 1: Eliminate Add the equations together to kill $y$ . $(5x + 3x) + (2y - 2y) = 20 + 12$ $8x = 32$

Step 2: Solve for x $x = 4$

Step 3: Solve for y Substitute $x = 4$ into equation (1). $5(4) + 2y = 20$ $20 + 2y = 20$ $2y = 0$ $y = 0$

Solution: $x = 4, y = 0$

What if they don’t match?

If you have $2x + 3y = 5$ and $x - y = 0$ , multiply the second equation by 2 (or 3) to make them match!

The Comparison: Which One To Choose?

Scenario	Recommended Method	Why?
$y = \dots$ or $x = \dots$	Substitution	No need to rearrange terms. Just plug and play.
$3x + 4y = 10 \dots$	Elimination	Fractions are messy. Elimination keeps numbers whole.
$y = 2x$ and $y = x + 3$	Equal Values	Set $2x = x + 3$ . It’s technically substitution!

❌ The #1 Mistake: The Negative Zone

Subtraction Errors

When you subtract equations, be careful with negative signs.

$3x - (-2x) = 5x$

Many students write $3x - 2x = x$ . Don’t do that. Put brackets around the second equation!

Summary checklist

Check for Isolation: Is $x$ or $y$ alone? Use Substitution.
Check for Matching: Do coefficients match? Use Elimination.
Check Signs: Are you adding or subtracting? Watch the negatives!
Check Answer: Plug your $x$ and $y$ back into both equations to verify.

Simultaneous Equations: The Complete Guide (Substitution vs Elimination)